Description and Analysis of Procedural and Declarative Knowledge

The foundation year, exactly as its name suggests forms the foundation for all mathematical learning and the content descriptors of all year levels can be traced back to this foundation year. If students do not develop a competent knowledge, understanding and ability to apply learning in this year their mathematical ability will be compromised throughout their schooling. This is evidenced with a learner I am currently working with who is a year 6 student but has gaps in knowledge going back to foundation understandings. This makes year 6 mathematics for this student almost impossible on many levels. It has been described by this student as like looking at a foreign language at times.
As we move through the currucilum looking at year levels and the relation to other year levels we have come to the conclusion that it cannot be stressed enough for Learning Managers to see the curriculum as a continuum and not an isolated yearly pocket. Students must understand the knowledge appropriate for each year level (and previous year levels) in order to progress successfully. It is the responsibility of the Learning Manager to ensure this understanding is met through individually looking at student’s needs and abilities and creating activities that will support individual students in progressing their learning.
The learning manager should use the curriculum as a continuum and look forward and back in years when planning any mathematics program to ensure students have the appropriate prior knowledge and that the learning occuring withiin the program leads to future content descriptors.

Foundation Year
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Establish understanding of the language and processes of counting by naming numbers in sequences, initially to and from 20, moving from any starting point (ACMNA001) Literacy Intercultural understanding Numeracy Aboriginal and Torres Strait Islander histories and cultures Asia and Australia´s engagement with Asia	Reading stories from other cultures featuring counting in sequence to assist students to recognise ways of counting in local languages and across cultures Identifying the number words in sequence, backwards and forwards, and reasoning with the number sequences, establishing the language on which subsequent counting experiences can be built Developing fluency with forwards and backwards counting in meaningful contexts, including stories and Rhymes Understanding that numbers are said in a particular order and there are patterns in the way we say them	P1 - Read stories featuring counting from other cultures. P2 – Recognize ways of counting in local languages. P3 – Recognise ways of counting across cultures. P4 - Recognise different cultures counting systems. P5 – Identify number words in sequence, backwards. P6 – Identify number words in sequence forwards. P7 – Identify and reason with number sequences backwards. P8 – Identify and reason with number sequences forwards. P9 - Numbers are said in a particular order and have different patterns. P10 - Use numbers from 0 to 20	D1 - Establish understanding of the language of counting by naming numbers in sequences, initially to and from 20, moving from any starting point. D2 – Establish understanding of the processes of counting by naming numbers in sequences to and from 20 moving from any starting point. D3 - Establish the language on which subsequent counting experiences can be built. D4 – Develop fluency with counting forwards in meaningful contexts including stories and rhymes. D5 – Develop fluency with counting backwards in meaningful context including stories and rhymes. D6 - Understand that numbers are said in a particular order and there are patterns in the way we say them
Analysis – prior knowledge required for understanding
(Assumed) Previous Knowledge: Kindergarten Children extend their understanding and engagement with symbols, pattern systems and numeracy concepts. Children’s learning indicator include: - Experimenting in play with mathematical tools, e.g. tape, measures, rulers, calculators, scales and measuring cups. - Manipulating and organizing objects to impose order and create groups, patterns, lines and sequences. - Using mathematical knowledge, concepts and language in their play. Teachers coordinate environments that: *Embed numeracy through spontaneous songs, rhymes, counting games and finger plays. *Include materials that support children’s developing numeracy understandings e.g. calendars, clocks, money, computers, mobile phones, number plates on cars, page numbers in books, signs and advertising, sports heroes and their own cultural icons (Ben 10, Hi 5) (C&K, 2011, pg 56). Students may have been to a C&K Kindergarten, childcare centre or day care facility before beginning their Prep year; the above prior knowledge may not have been learnt if the student has not attended. Students must have an understanding that numbers exist and that they have a use and purpose.

Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Connect number names, numerals and quantities, including zero, initially up to 10 and then beyond (ACMNA002) Literacy Intercultural understanding Numeracy Aboriginal and Torres Strait Islander histories and cultures Asia and Australia´s engagement with Asia	Understanding that each object must be counted only once, that the arrangement of objects does not affect how many there are, and that the last number counted answers the ‘how many’ question Using scenarios to help students recognise that other cultures count in a variety of ways, such as by placing one pebble in a bag to represent one object (for example to count the number of cattle).	P1 - Use scenarios and quality texts to help students recognise that other cultures count in a variety of ways. P2 - Connect number names, numerals and quantities, including zero, initially up to 10 and then beyond.	D1 - Understand each object must be counted only once, that the changing the arrangement of objects does not affect the amount, and that the last number counted answers the ‘how many’ question D2 - Understand that number names, numerals and quantities are connected.
Analysis – prior knowledge required for understanding
(Assumed) Previous Knowledge: Kindergarten Children extend their understanding and engagement with symbols, pattern systems and numeracy concepts. Children’s learning indicator include: - Experimenting in play with mathematical tools, e.g. tape, measures, rulers, calculators, scales and measuring cups. - Manipulating and organizing objects to impose order and create groups, patterns, lines and sequences. - Using mathematical knowledge, concepts and language in their play. Teachers coordinate environments that: *Embed numeracy through spontaneous songs, rhymes, counting games and finger plays. *Include materials that support children’s developing numeracy understandings e.g. calendars, clocks, money, computers, mobile phones, number plates on cars, page numbers in books, signs and advertising, sports heroes and their own cultural icons (Ben 10, Hi 5) (C&K, 2011, pg 56). Students may have been to a C&K Kindergarten, childcare centre or day care facility before beginning their Prep year; the above prior knowledge may not have been learnt if the student has not attended. Students may have a prior knowledge, which may have been modelled from their parents.
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Subitise small collections of objects (ACMNA003) Numeracy	Using subitising as the basis for ordering and comparing collections of numbers	P1 - Use subitising as the basis for ordering collections of numbers. P2 - Use subitising as the basis for comparing collections of numbers.	D1 - Understand how to subitise small collections of objects.
Analysis – prior knowledge required for understanding
(Assumed) Previous Knowledge: Kindergarten Children extend their understanding and engagement with symbols, pattern systems and numeracy concepts. Children’s learning indicator include: - Experimenting in play with mathematical tools, e.g. tape, measures, rulers, calculators, scales and measuring cups. - Manipulating and organizing objects to impose order and create groups, patterns, lines and sequences. - Using mathematical knowledge, concepts and language in their play. Teachers coordinate environments that: *Embed numeracy through spontaneous songs, rhymes, counting games and finger plays. *Include materials that support children’s developing numeracy understandings e.g. calendars, clocks, money, computers, mobile phones, number plates on cars, page numbers in books, signs and advertising, sports heroes and their own cultural icons (Ben 10, Hi 5) (C&K, 2011, pg 56). Students may have been to a C&K Kindergarten, childcare centre or day care facility before beginning their Prep year; the above prior knowledge may not have been learnt if the student has not attended. Students may have had prior experiences using subtilising through board games with parents and friends using dice or other props and objects.
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Compare, order and make correspondences between collections, initially to 20, and explain reasoning (ACMNA289) Literacy Intercultural understanding Personal and social capability Critical and creative thinking Aboriginal and Torres Strait Islander histories and cultures Asia and Australia´s engagement with Asia	Comparing and ordering items of like and unlike characteristics using the words ‘more’, ‘less’, ‘same as’ and ‘not the same as’ and giving reasons for these answers Understanding and using terms such as ‘first’ and ‘second’ to indicate ordinal position in a sequence. Using objects which are personally and culturally relevant to students	P1 - Compare correspondences between collections, initially to 20, and explain reasoning. P2 - Order correspondences between collections, initially to 20, and explain reasoning. P3 - Compare items of like and unlike characteristics using the words ‘more’, ‘less’, ‘same as’ and ‘not the same as’ and giving reasons for these answers. P4 - Order items of like and unlike characteristics using the words ‘more’, ‘less’, ‘same as’ and ‘not the same as’ and giving reasons for these answers. P5 - Use terms such as ‘first’ and ‘second’ to indicate ordinal position in a sequence. P6 - Use objects that are personally and culturally relevant to students.	D1 - Understand terms such as ‘first’ and ‘second’ to indicate ordinal position in a sequence. D2 - Recognise correspondences between collections, initially to 20. D3 - Recognise objects that are personally and culturally relevant to students. D4 - Be familiar with items of like and unlike characteristics using the words ‘more’, ‘less’, ‘same as’ and ‘not the same as’.
Analysis – prior knowledge required for understanding
(Assumed) Previous Knowledge: Kindergarten Children extend their understanding and engagement with symbols, pattern systems and numeracy concepts. Children’s learning indicator include: - Experimenting in play with mathematical tools, e.g. tape, measures, rulers, calculators, scales and measuring cups. - Manipulating and organizing objects to impose order and create groups, patterns, lines and sequences. - Using mathematical knowledge, concepts and language in their play. Teachers coordinate environments that: *Embed numeracy through spontaneous songs, rhymes, counting games and finger plays. *Include materials that support children’s developing numeracy understandings e.g. calendars, clocks, money, computers, mobile phones, number plates on cars, page numbers in books, signs and advertising, sports heroes and their own cultural icons (Ben 10, Hi 5) (C&K, 2011, pg 56). Students may have been to a C&K Kindergarten, childcare centre or day care facility before beginning their Prep year; the above prior knowledge may not have been learnt if the student has not attended. Students may have had experience with comparing and ordering by parents everyday conversations, using terminology such as more, less, first and second; teaching their child the order of numbers.
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Represent practical situations to model addition and sharing (ACMNA004) Critical and creative thinking Numeracy Aboriginal and Torres Strait Islander histories and cultures	Using a range of practical strategies for adding small groups of numbers, such as visual displays or concrete materials Using Aboriginal and Torres Strait Islander methods of adding, including spatial patterns and reasoning	P1 - Use a range of practical strategies for adding small groups of numbers, such as visual displays or concrete materials.	D1 - Understand how to use a range of practical strategies for adding small groups of numbers, such as visual displays or concrete materials.
Analysis – prior knowledge required for understanding
(Assumed) Previous Knowledge: Kindergarten Children extend their understanding and engagement with symbols, pattern systems and numeracy concepts. Children’s learning indicator include: - Experimenting in play with mathematical tools, e.g. tape, measures, rulers, calculators, scales and measuring cups. - Manipulating and organizing objects to impose order and create groups, patterns, lines and sequences. - Using mathematical knowledge, concepts and language in their play. Teachers coordinate environments that: *Embed numeracy through spontaneous songs, rhymes, counting games and finger plays. *Include materials that support children’s developing numeracy understandings e.g. calendars, clocks, money, computers, mobile phones, number plates on cars, page numbers in books, signs and advertising, sports heroes and their own cultural icons (Ben 10, Hi 5) (C&K, 2011, pg 56). Students may have been to a C&K Kindergarten, childcare centre or day care facility before beginning their Prep year; the above prior knowledge may not have been learnt if the student has not attended. Students may have gained a basic understanding of addition and sharing through parents. If the child has siblings they may be taught to share various materials in everyday life, giving them the basis to learn sharing and addition.

Year 1
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Develop confidence with number sequences to and from 100 by ones from any starting point. Skip count by twos, fives and tens starting from zero (ACMNA012) Literacy Intercultural understanding Numeracy Asia and Australia´s engagement with Asia	Using the popular Korean counting game (samyukgu) for skip counting Developing fluency with forwards and backwards counting in meaningful contexts such as circle games	P1 - Use the popular Korean counting game (samyukgu) for skip counting. P2 - Skip count by twos, fives and tens starting from zero. P3 - Partition numbers using different materials to make numbers between 0 to 100(using straws, paddle pop sticks, etc). P4 - Develop fluency with forwards counting in meaningful contexts such as circle games. P5 - Develop fluency with forwards counting in meaningful contexts such as circle games.	D1 - Develop an understanding of forwards counting in meaningful contexts. D1 - Develop an understanding of backwards counting in meaningful contexts. D3 - Develop confidence with number sequences to and from 100 by ones from any starting point.
Analysis – prior knowledge required for understanding
Previous Knowledge: - Foundation: Establish understanding of the language and processes of counting by naming numbers in sequences, initially to and from 20, moving from any starting point (ACMNA001) Students would have prior experience in counting to 20 from any starting point, they must understand the concept of skip counting, which would have been explored in foundation year with sharing.
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Recognise, model, read, write and order numbers to at least 100. Locate these numbers on a number line (ACMNA013) Literacy Critical and creative thinking	Modelling numbers with a range of material and images Identifying numbers that are represented on a number line and placing numbers on a prepared number line	P1 - Model counting numbers to at least 100. P2 - Read numbers to at least 100. P3 - Order numbers to at least 100 and locate these numbers on a number line. P4 - Model numbers with a range of materials and images. P5 - Identify numbers that are represented on a number line. P5 - Locate and place numbers on a prepared number line.	D1 - Recognise numbers to at least 100. D2 - Understand how to locate numbers to 100 on a number line.
Analysis – prior knowledge required for understanding
Previous Knowledge: - Foundation: Compare, order and make correspondences between collections, initially to 20, and explain reasoning (ACMNA289) Students need to have learnt to read, write and order numbers to 20 in foundation year, they would also need to understand number sequences to and from 100. Students would be introduced to a number line.
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Count collections to 100 by partitioning numbers using place value (ACMNA014) Literacy Critical and creative thinking	Understanding partitioning of numbers and the importance of grouping in tens Understanding two digit numbers as comprised of tens and ones/units	P1 - Count collections to 100 by partitioning numbers using place value. P2 - Use partitioning of numbers by grouping into tens. P3 - Use two digit numbers comprising of tens and ones/units.	D1 - Understand partitioning of numbers and the importance of grouping in tens. D2 - Understand two digit numbers as comprised of tens and ones/units.
Analysis – prior knowledge required for understanding
Previous Knowledge: - Foundation: Represent practical situations to model addition and sharing (ACMNA004) Students would have had little prior knowledge of partitioning, though they may be able to draw on their understanding of sharing to group and partition two digit numbers to 100.

Year 1

Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Represent and solve simple addition and subtraction problems using a range of strategies including counting on, partitioning and rearranging parts (ACMNA015) Literacy Critical and creative thinking Numeracy	Developing a range of mental strategies for addition and subtraction problems	P1 - Use a range of strategies including counting on, partitioning and rearranging parts to show and solve simple addition problems. P1 - Use a range of strategies including counting on, partitioning and rearranging parts to show and solve simple subtraction problems.	D1 - Develop a range of mental strategies for addition problems. D2 - Develop a range of mental strategies for subtraction problems.
Analysis – prior knowledge required for understanding
Previous Knowledge: - Foundation: Represent practical situations to model addition and sharing (ACMNA004) Students would understand the concept of addition and sharing and be able to use concrete materials to solve problems, they would be able to use one or more strategies to do this.
Year 2
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Investigate number sequences, initially those increasing and decreasing by twos, threes, fives and ten from any starting point, then moving to other sequences. (ACMNA026) Literacy Critical and creative thinking	Developing fluency and confidence with numbers and calculations by saying number sequences Recognising patterns in number sequences, such as adding 10 always results in the same final digit	P1 - Investigate number sequences, initially those increasing and decreasing by twos, threes, fives and ten from any starting point, then moving to other sequences. P2 - Developing fluency and confidence with numbers and calculations by saying number sequences P3 - Identify patterns in number sequences, such as adding 10 always results in the same final digit.	D1 - Develop confidence with numbers sequences and calculations. D2 - Recognise there are patterns in number sequences, such as adding 10 always results in the same final digit.
Analysis – prior knowledge required for understanding
Previous Knowledge: - Year 1: Develop confidence with number sequences to and from 100 by ones from any starting point. Skip count by twos, fives and tens starting from zero (ACMNA012) Count collections to 100 by partitioning numbers using place value (ACMNA014) Patterns and algebra sub strand: Investigate and describe patterns formed by skip counting and patterns with objects (ACMNA018) Students must have confidence to skip count by twos, fives and tens starting from zero, they will also understand why you would miss numbers in the sequence. Students prior knowledge of partitioning numbers would assist in identifying patterns in number sequences to find the same final digit. Developing fluency with numbers is important as it leads to future curriculum descriptors, if students do not have this fluency with numbers they are not likely to succeed later in maths.
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Recognise, model, represent and order numbers to at least 1000 (ACMNA027) Literacy Critical and creative thinking Numeracy	Recognising there are different ways of representing numbers and identifying patterns going beyond 100 Developing fluency with writing numbers in meaningful contexts	P1 - Recognise numbers to at least 1000. P2 - Model numbers to at least 1000. P3 - Order numbers to at least 1000. P4 - Recognise there are different ways of representing numbers and identifying patterns going beyond 100. P4 - Develop fluency with writing numbers in meaningful contexts.	D1 - Understand the order of numbers to at least 1000. D2 - Understand the different ways of representing numbers and patterns beyond 100.

Analysis – prior knowledge required for understanding
Previous Knowledge: - Foundation: Compare, order and make correspondences between collections, initially to 20, and explain reasoning (ACMNA289) - Year 1: Recognise, model, read, write and order numbers to at least 100. Locate these numbers on a number line (ACMNA013) Students must have prior understanding of numbers to at least 100. They must also be able to show different ways to represent and identify patterns.
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Group, partition and rearrange collections up to 1000 in hundreds, tens and ones to facilitate more efficient counting (ACMNA028) Literacy Intercultural understanding Critical and creative thinking Numeracy Aboriginal and Torres Strait Islander histories and cultures Asia and Australia´s engagement with Asia	Using an abacus to model and represent numbers Understanding three digit numbers as comprised of hundreds, tens and ones/units Demonstrating and using models such as linking blocks, sticks in bundles, place value blocks and Aboriginal bead strings and explaining reasoning	P1 - Group collections up to 1000 in hundreds, tens and ones to facilitate more efficient counting. P2 - Partition collections up to 1000 in hundreds, tens and ones to facilitate more efficient counting. P3 - Rearrange collections up to 1000 in hundreds, tens and ones to facilitate more efficient counting. P4 - Use an abacus to model and represent numbers. P5 - Demonstrate and use models such as linking blocks, sticks in bundles, place value blocks and Aboriginal bead strings and explaining reasoning.	D1 - Understand ways to group 1000 into hundreds, tens and ones to make more efficient counting. D2 - Understand three digit numbers as comprised of hundreds, tens and ones/units.
Analysis – prior knowledge required for understanding
Previous Knowledge: - Foundation: Compare, order and make correspondences between collections, initially to 20, and explain reasoning (ACMNA289) Represent practical situations to model addition and sharing (ACMNA004) - Year 1: Represent and solve simple addition and subtraction problems using a range of strategies including counting on, partitioning and rearranging parts (ACMNA015) Students must have prior understanding and recognise numbers to at least 1000. They must also be able to show different ways to represent and identify patterns. Students would also draw on prior learning of two digit numbers to inform learning of three digit numbers.
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Explore the connection between addition and subtraction (ACMNA029) Critical and creative thinking	Becoming fluent with partitioning numbers to understand the connection between addition and subtraction Using counting on to identify the missing element in an additive problem	P1 - Explore the connection between addition and subtraction. P2 - Becoming fluent with partitioning numbers to understand the connection between addition and subtraction. P3 - Use counting on to identify the missing element in an additive problem.	D1 - Become familiar with partitioning numbers to understand the connection between addition and subtraction. D2 - Understand the concept of counting on.
Analysis – prior knowledge required for understanding
Previous Knowledge: - Foundation: Represent practical situations to model addition and sharing (ACMNA004) - Year 1: Represent and solve simple addition and subtraction problems using a range of strategies including counting on, partitioning and rearranging parts (ACMNA015) Students have a clear understanding of the concept of addition and subtraction, this is needed to make a connection between the both.
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Solve simple addition and subtraction problems using a range of efficient mental and written strategies (ACMNA030) Literacy Critical and creative thinking Numeracy	Becoming fluent with a range of mental strategies for addition and subtraction problems, such as commutativity for addition, building to 10, doubles, 10 facts and adding 10 Modelling and representing simple additive situations using materials such as 10 frames, 20 frames and empty number lines	P1 - Become fluent with a range of mental strategies for addition problems; such as commutativity for addition, building to 10, doubles, 10 facts and adding 10. P2 - Become fluent with a range of mental strategies for subtraction problems. P3 - Solve simple addition and subtraction problems using a range of efficient mental and written strategies. P4 - Solve simple subtraction problems using a range of efficient mental and written strategies. P5 - Model simple additive situations using materials such as 10 frames, 20 frames and empty number lines. P5 - Represent simple additive situations using materials such as 10 frames, 20 frames and empty number lines.	D1 - Understand a range of efficient mental and written strategies to solve simple addition problems. D2 - Understand a range of efficient mental and written strategies to solve simple subtraction problems.

Analysis – prior knowledge required for understanding
Previous Knowledge: - Foundation: Represent practical situations to model addition and sharing (ACMNA004) - Year 1: Represent and solve simple addition and subtraction problems using a range of strategies including counting on, partitioning and rearranging parts (ACMNA015) Students must know how to solve simple addition and subtraction problems mentally and written, they need to use prior strategies and explore new strategies to solve problems.
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Recognise and represent multiplication as repeated addition, groups and arrays (ACMNA031) Literacy Critical and creative thinking Numeracy	Representing array problems with available materials and explaining reasoning Visualising a group of objects as a unit and using this to calculate the number of objects in several identical groups	P1 - Represent multiplication as repeated addition, groups and arrays. P2 - Represent array problems with available materials and explaining reasoning. P3 - Visualising a group of objects as a unit and using this to calculate the number of objects in several identical groups.	D1 - Recognise multiplication as repeated addition. D2 - Recognise multiplication as repeated groups. D3 - Recognise multiplication as repeated arrays.
Analysis – prior knowledge required for understanding
Previous Knowledge: - Foundation: Represent practical situations to model addition and sharing (ACMNA004) - Year 1: Represent and solve simple addition and subtraction problems using a range of strategies including counting on, partitioning and rearranging parts (ACMNA015) Students must know how to solve simple addition and subtraction problems mentally and written. Students need to understand that link between addition and multiplication, that it is repeated addition. Studeents must know how to solve simple addition and subtraction problems both mentally and in written form as they will need to solve more complex addition and subtraction problems in future mathematics descriptors.
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Recognise and represent division as grouping into equal sets and solve simple problems using these representations (ACMNA032) Critical and creative thinking Numeracy	Dividing the class or a collection of objects into equal sized groups Identifying the difference between dividing a set of objects into three equal groups and dividing the same set of objects into groups of three	P1 - Represent division as grouping into equal sets and solve simple problems using these representations. P2 - Dividing the class or a collection of objects into equal sized groups. P3 - Identify the difference between dividing a set of objects into three equal groups and dividing the same set of objects into groups of three.	D1 - Recognise division as grouping into equal sets and solve simple problems using these representations.
Analysis – prior knowledge required for understanding
Previous Knowledge: - Foundation: Represent practical situations to model addition and sharing (ACMNA004) - Year 1: Represent and solve simple addition and subtraction problems using a range of strategies including counting on, partitioning and rearranging parts (ACMNA015) Students must understand addition and subtraction concepts and realise that division is grouping into equal sets. This foundation knowledge of division sets the student up for future more complex division understanding.

Year 3
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Investigate the conditions required for a number to be odd or even and identify odd and even numbers (ACMNA051)	Identifying even numbers using skip counting by twos or by grouping even collections of objects in twos. Explaining why all numbers that end in the digits 0, 2, 4, 6 and 8 are even and that numbers ending in 1, 3, 5, 7 and 9 are odd.	Students will be able to:- P1: Use concrete materials to group items into sets of two P2: Skip count in twos from any number in both odd and even numbers. P3: Point to an odd number or an even number and tell the teacher what makes it odd or even	Students will know that:- D1: When all items in the group have been grouped in twos, it is an even number. D2: If some items have not been grouped into twos and one is over, the number is odd. D3: Skip counting can occur by starting from any number, not just zero.
Analysis – prior knowledge required for understanding
It is assumed that by the end of year 2 students would have had experiences in skip counting in odd and even numbers and would be able to recognise increasing and decreasing number sequences involving 2s, 3s and 5s. Students need to have used grouping concrete materials into sets to represent numbers in previous years. Students identify the missing element in a number sequence in order to do this students must have a sound understanding of the number sequence. Students count to and from 1000. They perform simple addition and subtraction calculations using a range of strategies. The knowledge learned about the properties of odd and even numbers, addition, subtraction and number sequences here leads to a more sophisticated knowledge in later years. Year 1: fluency in addition and patterning. Year 2: fluency with addition, patterning and counting on (ACMNA026). Year 2: recollection and representation of number facts (ACMNA031)
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Recognise, model, represent and order numbers to at least 10 000 (ACMNA052)	placing four digit numbers on a number line using an appropriate scale reproducing numbers in words using their numerical representations and vice versa	Students will be able to:-P1: Place given numbers onto a number line in digit form. P2: Circle a digit within a number to show either ones, tens, hundreds, thousands or ten thousands. P3: Place a given number on a number line or in order of largest to smallest or vice versa. P4: Transfer written numbers into digit form up to the tens of thousands. P5: Transfer numbers from digit form into written form	Students will know that:- D1: Numbers on a number line have a certain order. D2: each digit within a number has its own value. D3: Numbers and digits have an order and can be assending or descending but will always be in a certain order. D4: 25 thousands, 0 hundreds, 8 tens and 4 ones equals twenty five thousand and eighty four or 25,084 and vice versa.
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Recognise and explain the connection between addition and subtraction (ACMNA054)	demonstrating the connection between addition and subtraction using partitioning or by writing equivalent number sentences	Students will be able to:- P1: use concrete materials to show how 'more than' and 'less than' can be used in addition and subtraction. P2: use the parts known (addition) for materials and numbers to find an answer. example- Max has 3 cars and Jim has 5 cars, how many altogether? 3+5=? P3: use whole - part known (subtraction) using materials and numbers to find an answer. Example- Jodi has 8 chocolate eggs. She gave 3 to Hayley. How many eggs does Jodi have left? 8-3=?	Students will know that:- D1: Adding and subtracting can be done to solve problems D2: if two parts, either material or numericial, are known, they can be added together to find an answer. D3: if the whole number is known, either in concrete form or numerical, and also one other part, they can take the part known away from the whole to find an answer.
Analysis – prior knowledge required for understanding
It is assumed that by the end of year 2 students would have had experiences in simple addition and subtraction calculations using a range of strategies. Students must also have a deep understanding of number. Concrete materials will have been used from Prep through year 2 and this form of addition and subtraction will be familiar. Students will need to have an understanding of base 10 concepts in order to understand these descriptors. From Prep through to year 2 students will have been learning the base 10 concept this will aid learning in maths throughout the next few years of primary schooling. Prep - Represent practical situations to model addition and sharing (ACMNA004) Year 1 - developing a range of mental strategies for addition and subtraction problems Year 2. Explore the connection between addition and subtraction. (ACMNA029)
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Apply place value to partition, rearrange and regroup numbers to at least 10 000 to assist calculations and solve problems (ACMNA053)	recognising that 10 000 equals 10 thousands, 100 hundreds, 1000 tens and 10 000 ones l justifying choices about partitioning and regrouping numbers in terms of their usefulness for particular calculations	Students will be able to:- P1: Round numbers to 10 using different numbers and real life situations. P2: Choose a strategy like regrouping or partitioning to estimate an answer. P3: Use vertical placed algorithms to add up to 3 digit numbers P4: Use vertical placed algorithms to add up to 4 digit numbers P5: Use subtraction strategies to calculate the difference between numbers with up to three digits. P6: Use subtraction strategies to calculate the difference between numbers with up to 4 digits. P7: Subtract zero from any given number. P8: use a number expander to rename numbers; 3589 can be renamed as 35 hundreds and 89 ones.	Students will know that:- D1: Rounding numbers can assist with estimating the answers to problems in maths. D2: Choosing to partition or regroup numbers can aid with estimating an answer. D3: where you place the numbers in a algorithm will help get the correct answer. D4: number sentences can be in work form of in numerical form; Sydney is 2874kms from Rockhampton. Cape York is 2645kms from Rockhampton. Which town is closer to Rockhampton? What is the difference?/total? D5: when zero is added or subtracted from any number, that number stays the same.
Analysis – prior knowledge required for understandingStudents must understand place value in order to use it to partition numbers.It is also assumed that students have a sound number understanding in order to complete these descriptors. Students will demonstrate a procedural and declarative knowledge of partitioning, regrouping and algorithms by the end of this descriptor. Prior knowledge of subtraction and calculation must be sound in order to be competent here.
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Recall addition facts for single digit numbers and related subtraction facts to develop increasingly efficient mental strategies for computation (ACMNA055)	recognising that certain single digit number combinations always result in the same answer for addition and subtraction, and using this knowledge for addition and subtraction of larger numbers combining knowledge of addition and subtraction facts and partitioning to aid computation (for example 57 + 19 = 57 + 20 – 1)	Students will be able to:- P1: state all the number facts to 10; (1+9, 2+8, 3+7) and (10-5, 10-4, 10-3) in different combinations of addition and subtraction. P2: state the place value of a digit by its position in a number; the place value of 5 in 3587 is hundreds, or 500, (digit times the place value). P3: divide a number into parts by standard partitioning keeping the digits in their correct place; 8463 = 8000+400+60+3 P4: use non-standard partitioning to aid addition and subtraction; if 9+6 is 10+5 then the same can be done for larger numbers like 38+16 is 40+14. Both add up to 54 but digits have been taken from 16 to round up 38 to 40, making it easier to add the two together.	Students will know that:- D1: That number facts to ten can be used in many combinations D2: the different place value names and where they go in a given number. D3: partitioning numbers in standard and non-standard form can aid addition and subtraction of larger numbers.
Analysis – prior knowledge required for understanding
It is assumed that by the end of year 2 students would have had experiences in counting to and from 1000. They perform simple addition and subtraction calculations using a range of strategies. Students will have some knowledge and practise of using mental computations as well as concrete materials to assist with addition and subtraction. In order to recall addition facts students need a sound knowledge of addition facts. A sound knowledge of subtraction and of number is also required for this descriptor. Students must be able to partition numbers in a standard and non-standard way and have an understanding of place value. Year 1- Represent and solve simple addition and subtraction problems using a range of strategies including counting on, partitioning and rearranging parts (ACMNA015) Year 2- Group, partition and rearrange collections up to 1000 in hundreds, tens and ones to facilitate more efficient counting (ACMNA028) Having a good understanding of addition and subtraciton form the number and algebra strand will be a sound bases for other strands, in this instance it would align with Patterns and algebra in; Describe, continue, and create number patterns resulting from performing addition or subtraction (ACMNA060)
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Recall multiplication facts of two, three, five and ten and related division facts (ACMNA056)	establishing multiplication facts using number sequences	Students will be able to:- P1: multiply numbers in any order; 5x4=4x5. P2: recall the 2, 3, 5 and ten times tables. P3: relate the multiplication fact and the division fact for the 2, 3,5 and 10 numbers; 4x3=12, 3x4=12, 12÷4=3, 12÷3=4 P4: use an array to help with working out multiplication.	Students will know that:-D1: numbers used in multiplication can be used in any order; 34 45 x45 x34 and get the same result. D2: having the ability to recall timetables will assist with mental computation. D3: that multiplication and division are related as in P3 knowledge.

Analysis - prior knowledge required for understandingIn order to recall multiplication students must have a deep knowledge of the concept. Students must understand number sequences and skip counting in order to recall multiplication facts. Students will also need to understand repeated addition and why multiplication is a more efficient strategy (see issues and challenges page).

Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Represent and solve problems involving multiplication using efficient mental and written strategies and appropriate digital technologies (ACMNA057)	writing simple word problems in numerical form and vice versa using a calculator to check the solution and reasonableness of the answer	Students will be able to:- P1: students will be able to transfer a word problem into numerical from; Three buses are needed for the school swimming carnival to transport students. Each bus holds 46 students, how many students will be going to the carnival. 46x3=? P2: solve multiplication problems using distributive properties of multiplication spread over addition; 46x3=(40x3) + (6x3) = 120 + 18 = 138 P3: use a calculator to check their work.	Students will know that:- D1: a word problem can be translated into number form by reading and picking out relevant information. D2: the distributive property of multiplication over addition can be useful when multiplying larger numbers. D3: a calculator is not to replace mental computation, but for checking for reasonableness of the answer.
Analysis – prior knowledge required for understanding
It is assumed that by the end of year 2 students would have had experiences with mental computation and be aware of strategies used to work out multiplication and division algorithms. In order to be aware of and use these strategies students must have a sound knowledge of multiplication and division developed in previous years. Students will have had opportunities to use calculators and should be familiar with how to use them to check answers. Students will be familiar with repeated addition and will know this is not a replacement for multiplication problems but can be used alongside of multiplication problems. Students describe outcomes for everyday events using appropriate mathematical language.
Year 4
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Investigate and use the properties of odd and even numbers(ACMNA071)	using the four operations with pairs of odd or even numbers or one odd and one even number, then using the relationships established to check the accuracy of calculations.	Students will be able to:- P1: give explicit examples and explanations of an even number; 0,2,4,6,8,10,12….. P2: give explicit examples and explanations of an odd number; 1,3,5,7,9,11,13,15…. P3: give examples of the rules for adding, subtracting, multiplying and division of odd and even numbers.	Students will know that:- D1: a whole number is even if it is divisible by 2; 2,4,6,8… D2: an odd number is an integer that is not divisible by 2; 1.3.5.7.. D3: there are rules for odd and even algorithms which can assist with knowing what an answer might or might not be; E+E=E O+O=E O+E=O E+O=O E-E=E O-O=E O-E=O E-O=O ExE=E OxO=O OxE=E ExO=E E÷E=E O÷O=O O÷E=O/E with remainder E÷O=E with remainder
Analysis – prior knowledge required for understanding
It is assumed that by the end of year 3 students would have had experiences in:- recognising the connection between addition and subtraction and solve problems using efficient strategies for multiplication. Students count to and from 10 000. They classify numbers as either odd or even. They recall addition and multiplication facts for single digit numbers. They continue number patterns involving addition and subtraction. (ACMSP069) Students must be able to:- identifying odd and even numbers identifying sequences to assist with multiplication understanding the place value concept – times ten relationship between different places This knowledge must be soundly taught in order for students to succeed in later years when a more complex knowledge of odd and even properties is used.
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Recognise, represent and order numbers to at least tens of thousands (ACMNA072)	Reproducing five-digit numbers in words using their numerical representations, and vice versa.	Students will be able to:- P1: throw three dice, add two zeros and write five digit numbers. Order the numbers from greatest to smallest or vice versa. P2: transfer a number like 39853 onto a place value chart, placing all the numbers in their correct column. P3: reproduce the number 39853 from its numerical form onto words; thirty nine thousand, eight hundred and fifty three. P4: use an abacus to place a five digit number onto each place value then create a five digit number from reading beads on an abacus.	Students will know that:- D1: numbers can be ordered from smallest to largest or vice versa. D2: numbers with five digits are in the tens of thousands with each number having its own place. D3: numbers can be written in numerical form and in words. D4: numbers can be recognised in different forms other than words and numbers.
Analysis – prior knowledge required for understanding
It is assumed that by the end of year 3 students would have had experiences in:- Counting to and from 10 000 and be familiar with larger numbers. Identifying sequences and the order of numbers. Understanding the place value concept. A strong and flexible understanding of number is developed throughout all units in the year. This occurs through building on students’ place value understanding from Year 3, becoming fluent with addition and subtraction number facts, and applying computation strategies to increasingly sophisticated numbers. A strong and flexible understanding of number and place value is essential for later years in this sub-strand and also for use of fractions and decimals. Year 1- Recognise, model, read, write and order numbers to at least 100. Locate these numbers on a number line (ACMNA013) Year 2 - Recognise, model, represent and order numbers to at least 1000 (ACMNA027)
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Apply place value to partition, rearrange and regroup numbers to at least tens of thousands to assist calculations and solve problems (ACMNA073)	Recognising and demonstrating that the place value pattern is built on the operations of multiplication or division of tens	Students will be able to:- P1: use number expanders to reveal a number, decide how many tens it would take to divide or multiply to match another number in a different place value position; 7 hundreds are revealed how many tens need to be divided or multiplied to match 7 ones? P2: rearrange numbers to assist with calculations; 6+9+4=6+4+9= = 10+9=19 P3: partition numbers to assist with calculations; 5938 can be partitioned to 5000+900+30+8 P4: Regroup numbers to assist with algorithms; In subtraction 589-183 can be regrouped as 400+180+9 making the subtraction easier by taking the 180 away and 3 off the 9 leaving 406.	Students will know that:- D1: each number on a number line can be multiplied or divided by ten; 38495 the 9 represents 9 tens, then 49 tens, 849 tens, 3849 tens. The 5 is always 5 ones. D2: numbers can be rearranged but still have the same value in order to make the calculation easer. D3: standard and non-standard partitioning can be used as another method of calculation D4: regrouping numbers is standard partitioning which can be used in algorithms.
Analysis – prior knowledge required for understanding
It is assumed that by the end of year 3 students would have had experiences in recognising the connection between addition and subtraction and solve problems using efficient strategies for multiplication. Students count to and from 10 000. Prior to year three students must develop a sound knowledge and understanding of partitioning, rearranging and grouping numbers, this requires sound number knowledge and number sequencing knowledge alongside place value knowledge. Year 1 - Problem Solving in year 1 includes using materials to model authentic problems,and using familiar counting sequences to solve unfamiliar problems and discusing the reasonableness of the answer. Year 2 - Understanding includes connecting number calculations with counting sequences, partitioning and combining numbersflexibly, identifying and describing the relationship between addition and subtraction and between multiplication and division. Year 3 - Apply place value to partition, rearrange and regroup numbers to at least 10 000 to assist calculations and solve problems (ACMNA053) Being familar with place value system will assist students with negative numbers and where they are in the place value system. The fraction and decimal strand; Recognise that the place value system can be extended to tenths and hundredths. Make connections between fractions and decimal notation (ACMNA079) is an example of similarities and alignment.
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Investigate number sequences involving multiples of 3, 4, 6, 7, 8, and 9 (ACMNA074)	Recognising that number sequences can be extended indefinitely, and determining any patterns in the sequences	Students will be able to:- P1: communicate their understandings of the commutative properties of addition and multiplication; 4+5=5+4 and 3x7=7x3 and that this law does not apply to subtraction and division. P2: recall the fact that multiplication and division are linked by stating the following or similar; if 8x9=72, then 72÷9=8. P3: name some strategies for multiplication like; all multiples of 2 are even numbers ending in 0,2,4,6,8. Multiples of 3s, all the digits add up to 3,6 or 9. Doubling and doubling again is the same as multiplying by 4. A number can be divided by 4 if the last two digits are a multiple of 4. All multiples of 5 end with 5 or 0 and alternate as odd and even numbers. If a number is even and can be divided by 3 it is divisible by 6. If the digits in any number can be added to 9 it can be divided by 9; 585 because 5+8+5=18 and 1+8=9 so 585÷9=65	Students will know that:- D1: addition and multiplication can be ‘turnarounds’. Division and subtraction cannot. D2: when 2 numbers are multiplied, the product of that multiplication can then be divided by either of those numbers. D3: there are several strategies that can be used to help work out times tables and other algorithms.
Analysis – prior knowledge required for understanding
It is assumed that by the end of year 3 students would have had experiences in recognising the connection between addition and subtraction and solve problems using efficient strategies for multiplication. Students count to and from 10 000, in order to do this sound number knowledge must be developed alongside sequencing understanding and skip counting.
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Recall multiplication facts up to 10 × 10 and related division facts (ACMNA075)	Using known multiplication facts to calculate related division facts.	Students will be able to:- P1: communicate their understandings of the commutative properties of addition and multiplication; 4+5=5+4 and 3x7=7x3 and that this law does not apply to subtraction and division. P2: recall the fact that multiplication and division are linked by stating the following or similar; if 8x9=72, then 72÷9=8. P3: name some strategies for multiplication like; all multiples of 2 are even numbers ending in 0,2,4,6,8. Multiples of 3s, all the digits add up to 3,6 or 9. Doubling and doubling again is the same as multiplying by 4. A number can be divided by 4 if the last two digits are a multiple of 4. All multiples of 5 end with 5 or 0 and alternate as odd and even numbers. If a number is even and can be divided by 3 it is divisible by 6. If the digits in any number can be added to 9 it can be divided by 9; 585 because 5+8+5=18 and 1+8=9 so 585÷9=65	Students will know that: D1: addition and multiplication can be ‘turnarounds’. Division and subtraction cannot. D2: when 2 numbers are multiplied, the product of that multiplication can then be divided by either of those numbers. D3: there are several strategies that can be used to help work out times tables and other algorithms.
Analysis – prior knowledge required for understanding
It is assumed that by the end of year 3 students would have had experiences in recognising the connection between addition and subtraction and will be able to solve problems using efficient strategies for multiplication and division. Students will have used calculators to check answers and to assist with working out answers for larger algorithms. Students count to and from 10 000 and have a sound understanding of the concept of multiplication and how to successfully engage with it. At the end of year 2 students should have a sound understanding of partitioning and combining numbers flexibly, as well as identifying and describing the relationship between addition and subtraction and between multiplication and division. By the end of year 3, students should be fluent in recalling multiplication facts, this will assist with learning related division facts.
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Develop efficient mental and written strategies and use appropriate digital technologies for multiplication and for division where there is no remainder (ACMNA076)	using known facts and strategies, such as commutative, doubling and halving for multiplication, and connecting division to multiplication when there is no remainder	Students will be able to:- P1: show they can do multiplication and division algorithms in more than one way by choosing different strategies learned and show they have applied rules to suit. P2: use digital technology, namely a calculator, to check answers for algorithms.	Students will know that: D1: the various ways of approaching multiplication and division algorithms. D2: a calculator can be used to check answers and is a digital tool to assist with larger computations.
Analysis – prior knowledge required for understanding
It is assumed that by the end of year 3 students would have had experiences in recognising the connection between addition and subtraction and will be able to solve problems using efficient strategies for multiplication and division. Students will have used calculators to check answers and to assist with working out answers for larger algorithms. Students count to and from 10 000. Year 3 - students will have had experiences with reasoning with generalising from number properties and results of calculations, this will assist with mental computations.

Year 5

Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Identify and describe factors and multiples of whole numbers and use them to solve problems (ACMNA098)	exploring factors and multiples using number sequences using simple divisibility tests	Students will be able to:-P1: work out and give examples of missing factors. 9 the factors are 1,3,9, And 15 the factors are 1,3,5,15 Common factors are 1 and 3 P2: give examples of and describe a multiple. A number that can be multiplied by another number without a remainder:- 14 is a multiple of 2 and 7 P3: solve a word problem by using their knowledge of multiples and factors. Trent hung 15 shirts equally on each strand of the washing line. If there were 5 strands, how many shirts are on each?	Students will know that: D1: that a factor is a number that divides a larger number without any remainders. And that every number has at least two factors, the number it’s self and 1. D2: that common factors are factors that are exact divisors of two or more numbers. D3: a multiple is what results from multiplying a whole number by another whole number. 2x7=14 so 14 is the multiple of 2 and 7. D4: by using knowledge of multiples and factors, problems are solved easier. 15 shirts and 5 lines – there must be 3 shirts on each line because I know that 3x5=15
Analysis – prior knowledge required for understanding By the end of Year 4, students choose appropriate strategies for calculations involving multiplication and division. They recognise common equivalent fractions in familiar contexts and make connections between fraction and decimal notations up to two decimal places. They identify unknown quantities in number sentences. They describe number patterns resulting from multiplication. Year 4 - Students will have recalled multiplication facts of two, three, five and ten and related division facts (ACMNA056) - Represent and solve problems involving multiplication using efficient mental and written strategies and appropriate digital technologies (ACMNA057)
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Use estimation and rounding to check the reasonableness of answers to calculations (ACMNA099)	recognising the usefulness of estimation to check calculations applying mental strategies to estimate the result of calculations, such as estimating the cost of a supermarket trolley load	Students will be able to: P1: show their knowledge of making estimations to check a calculation Chloe has saved $227 to spend during her 4 week holiday to America. To make sure she doesn’t run out, what is the most Chloe can spend each week? P2: mentally calculate the total of a small shopping trolley by rounding and adding items in the cart.	Students will know that: D1: by their work showing reasonable examples of estimations and rounding. A solution to the Chloe problem could be; 227 can be rounded down to 200, the zero dropped making 20, 20÷4 =5, add the zero back on making $50 per week. Chloe could spend about $50 a week and not run out of money. D2: estimation can be used as an approximate calculation rather than just guessing, by using rounded numbers.
Analysis – prior knowledge required for understanding By the end of Year 4, students choose appropriate strategies for calculations involving multiplication and division. They recognise common equivalent fractions in familiar contexts and make connections between fraction and decimal notations up to two decimal places. They identify unknown quantities in number sentences. They describe number patterns resulting from multiplication. Students will also have knowledge and understanding from the use of concrete materials from prep through to year 4, this allows students to get a visual understanding of arrays and associating this with multiplication and division facts.
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Solve problems involving multiplication of large numbers by one or two digit numbers using efficient mental, written strategies and appropriate digital technologies (ACMNA100)	exploring techniques for multiplication such as the area model, the Italian lattice method or the partitioning of numbers applying the distributive law and using arrays to model multiplication and explain calculation strategies	Students will be able to:- P1: show on paper their understanding of multiplication strategies used when multiplying large numbers using one and two digit numbers. P2: explain the use of the distributive law when multiplying, either mentally or written using arrays as necessary. 6x(20+4)=(6x20) + (6x4)=144 6x 24 = 120 + 24=144 P3: Students will be able to show the use of a calculator to check answers.	Students will know that: D1: use various strategies to assist with multiplication; The area model which can lead on to the lattice model. D2: the use of the distributive law can assist with calculating multiplication algorithms involving larger numbers. D3: the use of a calculator can be one way of checking answers.
Analysis – prior knowledge required for understanding By the end of Year 4, students choose appropriate strategies for calculations involving multiplication and division. They recognise common equivalent fractions in familiar contexts and make connections between fraction and decimal notations up to two decimal places. They identify unknown quantities in number sentences. They describe number patterns resulting from multiplication. Year 4 experiences - Recall multiplication facts up to 10 × 10 and related division facts (ACMNA075) - Investigate number sequences involving multiples of 3, 4, 6, 7, 8, and 9 (ACMNA074)
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Solve problems involving division by a one digit number, including those that result in a remainder (ACMNA101)	using the fact that equivalent division calculations result if both numbers are divided by the same factor interpreting and representing the remainder in division calculations sensibly for the context	Students will be able to:- P1: use numbers from a number sentence to show their knowledge that when the numbers are multiplied or divided by a common factor, the multiplied or divided numbers are equivalent to the original numbers; Example 15÷5=45÷15 (15x3)÷(5x3)= 45÷15 45÷15=45÷15=3 P2: create a number sentence when the number has been divided into equal parts and an amount is left over, the remainder. 57÷9=6r3. P3: create a number sentence from a written problem where there will be a remainder. There are 17 cats needing accommodation, they can be housed 2 together. How many cat pens will be needed to house the 17 cats? 17÷2=8 remainder 1 One extra pen will be needed to accommodate the remaining cat, so 9 pens will be required altogether.	Students will know that: D1: common factors can be used to solve equivalent ‘missing number’ problems in a variety of ways. D2: the amount left over when a number has been divided into equal parts, the value of the remainders is always less than the divisor. 57÷9=6 remainder3 3 is smaller than 6. D3: a real life problem can be solved using facts know about division with remainders.
Analysis – prior knowledge required for understanding By the end of Year 4, students choose appropriate strategies for calculations involving multiplication and division. They recognise common equivalent fractions in familiar contexts and make connections between fraction and decimal notations up to two decimal places. They identify unknown quantities in number sentences. They describe number patterns resulting from multiplication. Year 4 experiences - Recall multiplication facts up to 10 × 10 and related division facts (ACMNA075) - Investigate number sequences involving multiples of 3, 4, 6, 7, 8, and 9 (ACMNA074)
Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Use efficient mental and written strategies and apply appropriate digital technologies to solve problems (ACMNA291)	using calculators to check the reasonableness of answers	Students will be able to:- P1: use a guess, check improve strategy to find an answer by using a sensible guess, writing down each step, then checking the guess with a calculator, and using the information to improve the guess. 783÷27(=29) Students offer a possible quotient, for example students might use rounding or a random number like 24. This is checked using a calculator 24x27=648 Not enough 783-648=135 24 was a reasoonalbe guess but is too low by 135 Using the information from above the student now improves the guess by knowing the number needed is above 24 but not too far away. They might try 26 26x27=702, almost there 27x27=729 29x27=783 spot on.	Students will know that: D1: That maths problems can be solved by using reasonable guesses as long as the guess is improved on and all stages are recorded. D2: calculators can be used to check guesses and that it is essential to show written evidence of the improvement on the guess.
Analysis – prior knowledge required for understanding By the end of Year 4, students choose appropriate strategies for calculations involving multiplication and division. They recognise common equivalent fractions in familiar contexts and make connections between fraction and decimal notations up to two decimal places. They identify unknown quantities in number sentences. They describe number patterns resulting from multiplication. Year 4 - Students will have recalled multiplication facts of two, three, five and ten and related division facts (ACMNA056) - Represent and solve problems involving multiplication using efficient mental and written strategies and appropriate digital technologies (ACMNA057)

Year 6

Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Identify and describe properties of prime, composite, square and triangular numbers (ACMNA122) Literacy Information and communication technology capability Critical and creative thinking	understanding that some numbers have special properties and that these properties can be used to solve problems representing composite numbers as a product of their prime factors and using this form to simplify calculations by cancelling common primes understanding that if a number is divisible by a composite number then it is also divisible by the prime factors of that number (for example 216 is divisible by 8 because the number represented by the last three digits is divisible by 8, and hence 216 is also divisible by 2 and 4)	P1 – Identify properties of prime, composite, square and triangular numbers. P2 – Describe properties of prime, composite, square and triangular numbers. P3 – represent composite numbers as a product of their prime factors. P4 – Use composite numbers to simplify calculations by cancelling prime numbers P5 – Use the properties of prime, composite, square and triangular numbers to solve problems.	D1 – Understand properties of prime, composite, square and triangular numbers. D2 – Understand that the properties of prime, composite, square and triangular numbers can be used to solve problems.
Analysis – prior knowledge required for understanding
Previous knowledge: Students must understand the properties of odd and even. This knowledge is especially required for prime numbers as students recognize the divisional powers of odd and even numbers in relation to their categorization into prime and composite numbers. Students need to understand how multiplication works in order to categorise a number as prime or composite as they will need to divide this number which is the process of reversing multiplication. In order to understand and work out triangle numbers students must be fluent with addition (begins in year 1), with patterning and counting on. This knowledge will continue to build and is crucial to working out triangular numbers as students count on to work out triangle numbers. Students must understand the concept of patterning numbers for triangle numbers as the pattern is counting on 2 then 3 then 4 then 4 etc. each time you count on you add an additional 1 to the previous number to move to the next triangle number. Triangle numbers are 3, 6, 10, 15 etc. Year 1: fluency in addition and patterning. Year 2: fluency with addition, patterning and counting on (ACMNA026). Year 2: recollection and representation of number facts (ACMNA031) Year 3: concepts of odd and even numbers. (ACMNA051). Recollection and representation of multiplication facts (ACMNA57). Year 4: Students begin to investigate the properties of odd and even numbers (ACMNA071). Year 5: students identify and describe factors and multiples of whole numbers (ACMNA098).

Year 6

Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers (ACMNA123) Literacy Information and communication technology capability Critical and creative thinking Numeracy	Applying strategies already developed for solving problems involving small numbers to those involving large numbers. Applying a range of strategies to solve realistic problems and commenting on the efficiency of different strategies.	P1 – select efficient mental strategies to solve problems involving all four operations with whole numbers P2 – select efficient written strategies to solve problems involving all four operations with whole numbers P3 – select appropriate digital technologies to solve problems involving all four operations with whole numbers. P4 – apply known strategies for solving problems with small numbers to large numbers. P5 – applying a range of strategies to solve realistic problems. P6 – comment on the effectiveness of various known and used strategies.	D1 – understand efficient mental and written strategies to solve problems with all four operations with whole numbers.. D2 – understand digital technologies to solve problems with all four operations with whole numbers. D3 – understand how to apply strategies already developed for solving problems involving small numbers to those involving large numbers. D4 – Understand a range of strategies to solve problems D5 – understand how to evaluate the efficiency of different strategies.
Analysis – prior knowledge required for understanding
Previous knowledge: Understand and competently use problem solving strategies. Understand and competently use operations. Understand number value. Content descriptors: Year 2 – the addition and subtraction problems are introduced to students. A sound understanding and application knowledge is needed here before students can move to more complex problems. Year 3 – students make a connection between addition and subtraction. Students develop mental strategies for recalling addition and related subtraction facts. Students use mental and written strategies to solve multiplication problems. Year 4 – students must recall multiplication facts and related division facts up to 10x10. Students must also develop efficient mental and written and technological strategies to solve multiplication and division facts. Year 5 – students are introduced to estimation to obtain a reasonable answer to calculations. Students solve multiplication problems of one or two digit numbers. Students solve division problems of one number including remainders. Students use mental and written strategies to solve problems.

Year 6

Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Investigate everyday situations that use integers. Locate and represent these numbers on a number line (ACMNA124) Literacy Critical and creative thinking Numeracy	Understanding that integers are ...3, 2, 1, 0, 1, 2, 3,..... l solving everyday additive problems using a number line investigating everyday situations that use integers, such as temperatures using number lines to position and order integers around zero solving everyday additive problems involving positive and negative integers without developing formal rules for the operations (for example using a number line and counting to find the resulting outside temperature if it is 5°C at 7pm and drops by 8°C overnight)	P1 – investigate everyday situations that use integers. P2 – locate and represent integers on a number line. P3 – solve everyday additive problems using a number line. P4 – investigate everyday situations that use integers such as temperatures. P5 – Use different number lines to position and order integers around zero. P6 – solve everyday additive problems involving positive and negative integers using a number line and counting.	D1 – understand that integers are whole negative and positive numbers. D2 – understand how integers are used in everyday situations. D3 – understand number lines D4 – understand problem solving
Analysis – prior knowledge required for understanding
Previous knowledge: Students must have an understanding and competency of integers and the terms positive and negative, an understanding of numbers lines and competency with use of numbers lines. Students must also have competency with addition, subtraction, multiplication and division in order to use integers in everyday situations. Students must have some exposure to and understanding of temperatures and their use. In order to become competent in this content descriptor students must have a thorough understanding and practical knowledge of numbers, their uses, their relationships and their properties. Many content descriptors in prior years lead to this knowledge. Content descriptors: Year 1: Knowledge of number sequences (ACMNA012). Year 2: Investigate number sequences (ACMNA026). Represent and organize numbers (ACMNA027). Grouping, partitioning and rearranging numbers (ACMNA028). Year 3: Understand and recognize odd and even numbers (ACMNA051). Representing, ordering and modeling numbers (ACMNA052). Year 4: Properties of odd and even numbers (ACMNA071). Recognising, representing and ordering numbers (ACMNA072).Investigating number sequences (ACMNA074). Year 5: Estimation and rounding to check reasonableness of answers (ACMNA099). Future knowledge: This knowledge will be used in future years when students look at operations in year 8.

Year7

Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Investigate index notation and represent whole numbers as products of powers of prime numbers (ACMNA149) Literacy Critical and creative thinking	defining and comparing prime and composite numbers and explaining the difference between them applying knowledge of factors to strategies for expressing whole numbers as products of powers of prime factors, such as repeated division by prime factors or creating factor trees solving problems involving lowest common multiples and greatest common divisors (highest common factors) for pairs of whole numbers by comparing their prime factorisation	P1 – Investigate index notation P2 – represent whole numbers as products of powers of prime numbers. P3 – Define and composite numbers P4 – Compare prime and composite numbers P5 – Explain the difference between prime and composite numbers. P6 – Apply knowledge of factors to strategies for expressing whole numbers as products of prime factors. P7 – use repeated division by prime factors as a strategy to express whole numbers as products of powers of prime factors. P8 – create factor tree. P9 – Solve problems using lowest common multiples. P10 – solve problems using greatest common divisors P11 – Use comparison of prime factors to solve problems using lowest common multiples and greatest common divisors.	D1 – Understand index notation D2 – understand products of powers of prime numbers. D3 – understand representation of whole numbers as products of powers of prime numbers. D4 – Understand prime and composite numbers and the different between them. D5 – Understand whole numbers as products of powers of prime numbers. D6 – Understand repeated division. D7 – Understand lowest common multiples. D8 – Understand greatest common divisors.
Analysis – prior knowledge required for understanding
Previous Knowledge: Students require knowledge of factors including prime and composite numbers and multiples. A thorough knowledge of multiplication is also essential and students should have acquired this through previous years of study. Students will also need to understand multiples, factors and prime numbers to use and understand index notation. Content descriptors: Year 2: Use of multiplication (ACMNA031). Year 3: Multiplication facts of two, three, five and ten (ACMNA056). Year 4: Recall multiplication facts (ACMNA075). Efficient mental and written strategies (ACMNA076). Year 5: Identify and describe factors and multiples (ACMNA098). Solve problems using multiplication (ACMNA100). Use of efficient mental and written strategies (ACMNA291). Year 6: Use of square and triangle numbers (ACMNA122). Future Knowledge: Students will use this knowledge to study index notation further in year 8.

Year 7

Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Investigate and use square roots of perfect square numbers (ACMNA150) Critical and creative thinking	investigating square numbers such as 25 and 36 and developing square-root notation investigating between which two whole numbers a square root lies	P1 – Investigate square roots of perfect numbers. P2 – use square roots of perfect numbers. P3 – develop square root notation. P4 – investigate between which two whole numbers a square root lies.	D1 – understand a square number D2 –understand square root notation D3 – understand why square roots lie between two whole numbers.
Analysis – prior knowledge required for understanding
Previous Knowledge: A complete understanding of numbers, values, properties and uses; this must be developed in the early years of learning. Students must have a complete understanding and knowledge of square and triangle numbers in order to investigate and use square roots of perfect square numbers. This content descriptor uses knowledge from almost all previous year levels. Content descriptors: Year 1: Number sequences, ordering numbers, counting and partitioning (ACMNA012. ACMNA013, ACMNA014) Year 2: Recognize, represent and order numbers (ACMNA027). Grouping and partitioning numbers (ACMNA028). Year 3: Properties of odd and even numbers (ACMNA051). Recognizing, representing and ordering numbers (ACMNA052). Place value, partitioning, rearranging and regrouping numbers (ACMNA053). Year 4: Properties of odd and even numbers (ACMNA071). Recognizing, representing and ordering numbers (ACMNA072). Place value, partitioning, rearranging and regrouping numbers (ACMNA073). Year 5: Factors and multiples (ACMNA098). Year 6: Square numbers, triangle numbers, prime and composite numbers (ACMNA122).

Year 7

Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Apply the associative, commutative and distributive laws to aid mental and written computation (ACMNA151) Critical and creative thinking	understanding that arithmetic laws are powerful ways of describing and simplifying calculations	P1 – apply the associate law to aid mental and written computation P2 – apply the commutative law to aid mental and written computation. P3 – apply the distributive laws to aid mental and written computation. P4 – use arithmetic laws to describe and simplify calculations.	D1 – understand associate laws D2 – understand commutative laws. D3 – understand distributive laws D4 – understand mental and written computation D5 – understand how arithmetic laws are powerful ways of describing and simplifying calculations.
Analysis – prior knowledge required for understanding
Previous Knowledge: Students must understand calculations, mental and written computation and have an understanding of various laws of mathematics. An understanding and use of estimation requires mental computation. Content descriptors: Year 1: Solve simple addition and subtraction problems using a range of strategies (ACMNA015). Year 2: Explore connection between addition and subtraction (ACMNA029). Solve simple addition and subtraction problems using efficient strategies (ACMNA030). Multiplication as repeated addition groups and arrays (ACMNA031). Year 3: recognize and explain the connection between addition and subtraction (ACMNA054). Multiplication facts (ACMNA056). Problem solving (ACMNA057). Year 4: Multiplication facts (ACMNA075). Mental and written strategies for multiplatinum and division (ACMNA076). Year 5: Use of estimation and rounding (ACMNA099). Solving problems and use of efficient mental and written strategies (ACMNA101), ACMNA291). Year 6: written and mental problem solving (ACMNA123).

Year 7

Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Compare, order, add and subtract integers (ACMNA280) Critical and creative thinking Numeracy		P1 – compare integers P2 – order integers P3 – subtract integers	D1 – know how to compare integers D2 –know how to order integers D3 – know how to subtract integers
Analysis – prior knowledge required for understanding
Previous Knowledge: Students must understand integers and their properties and understand addition and subtraction in order to compare, order, add and subtract integers. The properties of numbers are investigated in different ways from year 1 through to year 8. Content descriptors: Year 1: Confidence with number sequences starts in year 1 (ACMNA012). Students learn numbers, number lines and counting. (ACMNA013, ACMNA014). Year 2: Investigating number sequences, recognizing, modelling and ordering numbers is taught in year 2. (ACMNA026, ACMNA027). Exploring the connection between addition and subtraction, solve simple addition and subtraction problems (ACMNA029, ACMNA030). Year 3: Recognise and explain the connection between addition and subtraction (ACMNA054). Year 4: Recognise, represent and order numbers (ACMNA072). Investigate number sequences (ACMNA074). Year 5: use of efficient mental and written strategies to solve problems (ACMNA291). Year 6: Properties of prime numbers (ACMNA122). Select and apply efficient mental and written strategies to solve problems with four operations (ACMNA123). Investigate everyday situations that use integers (ACMNA124). Future Knowledge: students will use this knowledge to further study integers and index notation in year 8 and beyond.

Year 8

Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Use index notation with numbers to establish the index laws with positive integral indices and the zero index (ACMNA182) Critical and creative thinking	evaluating numbers expressed as powers of positive integers	P1 – use index notation with numbers P2 – use index notation to establish the index laws P3 – establish index laws with positive integral indices P4 – establish index laws and the zero index. P5 – evaluate numbers expressed as powers of positive integers	D1 – understand index notation. D2 – understand index laws D3 – understand positive integral indices D4 – understand the zero index. D5 – understand how to express numbers as powers of positive integers.
Analysis – prior knowledge required for understanding
Previous Knowledge: In order to use index notation with numbers students must first understand integers and their properties including positive and negative. In order to understand the zero index students must understand zero as an integer. Students must also have a sound knowledge of multiplication, powers of numbers place value and number laws. Content descriptors: Year 1: Confidence with number sequences starts in year 1 (ACMNA012). Students learn numbers, number lines and counting. (ACMNA013, ACMNA014). Year 2: Investigating number sequences, recognizing, modelling and ordering numbers is taught in year 2. (ACMNA026, ACMNA027). Exploring the connection between addition and subtraction, solve simple addition and subtraction problems (ACMNA029, ACMNA030). Year 3: Properties of odd and even numbers (ACMNA051). Model, order, represent and order numbers (ACMNA052). Recall multiplication facts (ACMNA056). Year 4: Investigate properties of odd and eve numbers (ACMNA071). Recognise, represent and order numbers (ACMNA072). Investigate number sequences (ACMNA074). Year 5: Identify and describe factors and multiples (ACMNA098). Use of multiplication, mental and written strategies to solve problems (ACMNA100, ACMNA291). Year 6: Investigate everyday situations that use integers (ACMNA124). Use integers in every day situations (ACMNA124).

Year 8

Content Descriptor	Elaboration	Procedural Knowledge	Declarative Knowledge
Carry out the four operations with rational numbers and integers, using efficient mental and written strategies and appropriate digital technologies (ACMNA183) Information and communication technology capability Critical and creative thinking	using patterns to assist in finding rules for the multiplication and division of integers using the number line to develop strategies for adding and subtracting rational numbers	P1 – carry out the four operations with rational numbers P2 – carry out the four operations with integers. P3 – use efficient mental strategies to carry out the four operations P4 – use efficient written strategies to carry out the four operations P5 – use appropriate digital technologies to carry out the four operations P6 – use patterns to assist with finding rules for the multiplication and division of integers. P7 – use the number line to develop strategies for adding and subtracting rational numbers.	D1- know the four operations in relation to rational numbers D2 – know the four operations in relation to integers D3 – know a range of mental, written and digital technologies strategies. D4 – understand patterns D5 – know how to apply patters to develop rules. D6 – understand what a rule is in mathematical terms. D7 – understand the multiplication and division of integers. D8 – understand the use of a number line and how to apply it
Analysis – prior knowledge required for understanding
Previous knowledge: Use and understanding of the four operations, use of rules within these operations and the use of a number line to represent and use numbers. Content descriptors: Year 1: Confidence with number sequences (ACMNA012). Recognise, model, read, write and order numbers, counting collections using partitioning, represent and solve simple addition and subtraction problems (ACMNA013, ACMNA014 and ACMNA015) Year 2: Explore the connection between addition and subtraction and use these to solve problems (ACMNA029, ACMNA030). Recognise and represent multiplication as repeated addition, groups and arrays (ACMNA031). Year 3: Recognise and explain the connection between addition and subtraction (ACMNA054). Represent and use multiplication to solve problems (ACMNA057). Year 4: recognising, representing and ordering numbers (ACMNA071). Multiplication facts and use of mental and written strategies to solve problems (ACMNA075, ACMNA076) Year 5: Use factors and multiples of whole numbers to solve problems (ACMNA098). Use of estimation to check answers (ACMNA099). Solve problems using multiplication and division using efficient mental and written strategies and appropriate digital technologies (ACMNA100, ACMNA101 and AMCNA291). Year 6: Properties of prime, composite, square and triangular numbers (ACMNA122). Select and apply mental and written strategies and appropriate digital technologies (ACMNA123). Use of integers in everyday situations (ACMNA124). Year 7: Application of commutative and distributive laws 9ACMNA151).